tailieunhanh - Đề tài " The sharp quantitative isoperimetric inequality "

A quantitative sharp form of the classical isoperimetric inequality is proved, thus giving a positive answer to a conjecture by Hall. 1. Introduction The classical isoperimetric inequality states that if E is a Borel set in Rn , n ≥ 2, with finite Lebesgue measure |E|, then the ball with the same volume has a lower perimeter, or, equivalently, that () 1/n nωn |E|(n−1)/n ≤ P (E) . Here P (E) denotes the distributional perimeter of E (which coincides with the classical (n − 1)-dimensional measure of ∂E when E has a smooth boundary) and ωn is the measure of the. | Annals of Mathematics The sharp quantitative isoperimetric inequality By N. Fusco F. Maggi and A. Pratelli Annals of Mathematics 168 2008 941-980 The sharp quantitative isoperimetric inequality By N. Fusco F. Maggi and A. Pratelli Abstract A quantitative sharp form of the classical isoperimetric inequality is proved thus giving a positive answer to a conjecture by Hall. 1. Introduction The classical isoperimetric inequality states that if E is a Borel set in Rn n 2 with finite Lebesgue measure E then the ball with the same volume has a lower perimeter or equivalently that n n n En I n P E Here P E denotes the distributional perimeter of E which coincides with the classical n 1 -dimensional measure of @E when E has a smooth boundary and n is the measure of the unit ball B in Rn. It is also well known that equality holds in if and only if E is a ball. The history of the various proofs and different formulations of the isoperimetric inequality is definitely a very long and complex one. Therefore we shall not even attempt to sketch it here but we refer the reader to the many review books and papers . 3 18 5 21 7 13 available on the subject and to the original paper by De Giorgi 8 see 9 for the English translation where was proved for the first time in the general framework of sets of finite perimeter. In this paper we prove a quantitative version of the isoperimetric inequality. Inequalities of this kind have been named by Osserman 19 Bonnesen type inequalities following the results proved in the plane by Bonnesen in 1924 see 4 and also 2 . More precisely Osserman calls in this way any inequality of the form A E P E 2 - IM valid for smooth sets E in the plane R2 where the quantity A E has the following three properties i A E is nonnegative ii A E vanishes only when E is a ball iii A E is a suitable measure of the asymmetry of E. 942 N. FUSCO F. MAGGI AND A. PRATELLI In particular any Bonnesen inequality implies the isoperimetric inequality as well as the

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